Contractions and applications of crystal skeletons: Young quasisymmetric and Stanley symmetric functions

Abstract

The character of a connected $\mathfrak{sl}_n$-crystal is a Schur polynomial; the crystal can be further decomposed into quasicrystals, whose characters are the Gessel quasisymmetric functions. Crystal skeletons are obtained by contracting quasicrystals within crystal graphs. They generalize dual equivalence graphs, and can be used to prove the Schur expansion of a symmetric function when the quasisymmetric expansion is known. In this paper, we show that the crystal skeleton can be tiled further into components which we call quasicrystal skeletons, whose characters are Young quasisymmetric Schur functions. We characterize which edges in the crystal skeleton move between quasicrystal skeleton components. Contracting the quasicrystal skeleton components yields Bruhat order. We illustrate how these tools can be applied to symmetric functions by analyzing the Stanley symmetric functions.